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Q. 12.5

A plane layer of gray medium with thickness D and constant properties is initially at uniform temperature $T_0$. It has attenuation coefficient β, density ρ, and specific heat $c_υ.$ At time t = 0 the layer is subjected to a very cold environment. Consider only radiation transfer and obtain the transient temperature of the layer by using the emission approximation.

Verified Solution

From the results of Example 12.4, the instantaneous transient heat flux emerging from both boundaries of the layer is $q(t=)4\beta \sigma T^4(t)D$.The energy equation for the layer then becomes $ρc_υdT /dt=-4\beta \sigma T^4(t),$or, in dimensionless form,$d\vartheta /\overline{dt} =-4\vartheta ^4(\overline{t} )$ where $ϑ=T/T_0$ and $\overline{t}=(\beta \sigma T_0^3/\rho c_v)t.$ Integrating with the condition that ϑ = 1 at $\overline{t}=0$ gives the transient uniform temperature throughout the layer for the emission approximation as

$\vartheta (\overline{t} )=\frac{1}{(1+2\overline{t})^{1/3}}$